Grapham User Guide
Contents
1. June 2007 URL http www gnu org licenses gpl 3 0 standalone html 10 H Haario E Saksman and J Tamminen An adaptive Metropolis algorithm Bernoulli 7 2 223 242 2001 11 F V Jensen Bayesian Networks and Decision Graphs Springer Verlag New York 2001 ISBN 0 387 95259 4 12 K P Murphy Dynamic Bayesian Networks Representation Inference and Learning PhD thesis University of California Berkeley 2002 URL http www ai mit edu murphyk mypapers html 13 G O Roberts and J S Rosenthal Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms J Appl Probab 44 2 458 475 2007 14 G O Roberts and J S Rosenthal Examples of adaptive MCMC J Comput Graph Statist 18 2 349 367 2009 15 M Vihola Robust adaptive Metropolis algorithm with coerced acceptance rate Statist Comput 22 5 997 1008 2012 A The Algorithms The general form of the implemented algorithms as well as several adjustable param eters are described in Section A 1 The different initialisation strategies are described in Section A 2 If several sampling blocks are used a sweep update is performed i e they are updated sequentially and the algorithms described below are applied to each of the blocks separately The order in which the blocks are updated is static and does not change from iteration to another 16 A 1 The Implemented Adaptive Algorithms All but one of the algorithms implemented in2. 0 double nsqx c sqrt x 0 x 0 x 1 x 1 return nsqx nsqx log 1 cos 5 x 0 1 cos 5 x 1 This example among with some others can be found in the directory clib under the Grapham main directory One can test it once Grapham is compiled as follows make clib grapham clib circular_target lua The C functions can use the standard distributions defined in Grapham by including grapham_math h as in the example above The names of the functions are however slightly different That is the function d_beta must be called instead of dbeta For 15 more information the function prototypes with brief explanation can be found in the file grapham_math h References 1 The BUGS project URL http www mrc bsu cam ac uk bugs 2 JAGS just another Gibbs sampler URL http calvin iarc fr martyn software jags 3 The programming language Lua URL http www lua org 4 Netlib repository URL http netlib org 5 SIMD oriented fast Mersenne twister URL http www math sci hiroshima u ac jp m mat MT SF 6 C Andrieu and J Thoms A tutorial on adaptive MCMC Statist Comput 18 4 343 373 Dec 2008 7 Y F Atchad and J S Rosenthal On adaptive Markov chain Monte Carlo algorithms Bernoulli 11 5 815 828 2005 8 L Carvalho Numeric Lua project page 2005 URL http luaforge net projects numlua Visited 11th June 2009 9 Free Software Foundation GNU general public license version 3
3. 20 and let Z x reject with probability 1 a If one uses delayed rejection then P x is a kernel determined by a two phase algorithm The first phase is as described above but if the first proposal Y is rejected then another independent proposal Y gt is generated from the a Gaussian distribution with mean x and covariance ps where p gt 0 typically 0 lt p lt 1 is the parameter para dr This second proposal is accepted with probability nin 1 adaz N m x q Ya3x s 1 a X o J in which case Z Y2 and otherwise Z x Note the acceptance probability that is used by h is the one from the first proposal 17 The algorithm para algorithm aswam adaptive scaling within AM im plements all 1 4 In the case of AM i e para algorithm am the scaling is not adapted i e h 0 a 8 The option para algorithm asm Adaptive scaling Metropolis implements only 1 and 4 while the choice para algorithm metropolis i e a standard non adaptive Metropolis algorithm implementing only 1 The Rao Blackwellised versions of AM para algorithm rbam and para algorithm rbaswam correspond to para algorithm am and para algorithm ams respectively They implement instead of 2 and 3 the following recursions Mn 1 1 Nn 1 Mn HNn 1 o Xn Yn 1 Yn 1 F 1 O Xn Yn 1 Xn 6 Sn 1 1 Nn 1 Sn HNn 1 Xn Yn1 Yat Mn Yn41 M 1 Xn n 1 Xn
4. compile Grapham tar zxf grapham X Y tar gz cd grapham make If the above fails you must edit the file Makefile in before issuing make For example the line LUA_PATH GRAPHAM_ROOT 1lua 5 1 4 indicates that the compiled Lua package is found in the directory lua 5 1 4 under the Grapham main directory If you wish to use the dSFMT Mersenne Twister random number generator recom mended or the Numeric Lua package you must download the source packages of them and edit Makefile in to include the paths where dSFMT and or Numeric Lua are installed For example DSFMT_PATH GRAPHAM_ROOT dSFMT src 2 0 NUMLUA_PATH GRAPHAM_ROOT numlua would work if SFMT and Numeric Lua source packages are extracted and compiled in the directories dSFMT src 2 0 and numlua under the Grapham main directory 3 Running Grapham Having downloaded and compiled Grapham you can use it by invoking grapham under the Grapham main directory For example you can simulate the Gamma distribution by grapham models gamma_test lua Total dimension 1 sampling 1 with AM With Lua calls Results after 500000 samples 10000 burn in Functional average 0 005492 0 120116 Acceptance rate 40 06 Sampling CPU time 1 22s 2 40us sample 2 40us sample dim When you have succesfully run Grapham for the first time you can take a look inside the file models gamma_test lua The specifications in the file are explained in the following
5. gt 0 dinvgamma a Inverse gamma with shape amp gt 0 and scale B gt 0 dinvchi2 v Inverse chi squared with v gt 0 degrees of freedom dlaplace u b Laplace with mean u and scale inverse rate b gt 0 dlevy c L vy with scale c gt 0 dlogistic u s Logistic with location u and scale s gt 0 dlognorm m v Log normal with mean m and variance v gt 0 of log x dnorm m v Normal with mean m and variance v gt Q dpareto Xm k Pareto with location xm gt 0 and shape k gt 0 Function Parameters Description drayleigh s Rayleigh with scale s gt 0 dstudent v Student s t distribution with v gt 0 degrees of free dom There is also a uniform density function duniform returning zero regardless of the input arguments Note that this does not determine a proper distribution The discrete distributions available are Function Parameters Description dbinom n p Binomial with n gt 0 trials and success probability 0 lt p lt l dnbinom r p Negative binomial with parameters r gt 0 and 0 lt p lt l dpoisson Poisson with rate A gt 0 The implemented multivariate distributions are Function Parameters Description dmvnorm U Multivariate Gaussian distribution with mean u and covariance matrix dmvstudent wu n Multivariate t distribution with location u scale and n gt 0 degrees of freedom dwishart Vin Wishart distribution with positive definite scale ma trix V and n gt 0 degrees of
6. the model That is a subset of the nodes in the model are duplicated and considered as in dependent repetitions The actual model specification of Grapham does not provide way to describe this kind of model but it provides the function repeat_block for actually creating these repeated blocks of variables in the table model The function repeat_block can be called in different ways The first argument always contains a list of variable names that are in the block to be replicated If the second argument is a number N the block is repeated N times The other case is that the following arguments are tables determining the values that the variables have An example of the latter would be repeat_block y x 2 410 9 0 7 0 6 0 8 This command would create variables y1 having value 0 9 and y2 with value 0 7 Similarly x1 and x2 have values 0 6 and 0 8 respectively But the variables z1 and z2 are left unknown One can also create chains of random variables appearing most importantly in dy namic models This is achieved by the postfix after the name of the parent vari able For example the following code creates a Markov chain Xoo with Xp 0 co and Xx41 Xk N Xx 1 for k gt 0 which means that X is the value of a standard Brownian motion at time k parents x Way y density dnorm J repeat_block x 10 This example can be found in the file models bm lua 5 Functional
7. Grapham User Guide Matti Vihola May 24 2013 1 Introduction Grapham is free experimental software It comes without any warranty See the GNU General Public License 9 for details 1 1 Objective The main objective of Grapham is to produce numerical estimates of integrals of the type TEOLO where f is a function of interest on the space X which is typically Rf but can also have integer valued components 1 The term graphical model in this setting simply means that the target density 7 or the model is determined by a collection of conditional probability densities For example suppose X R3 and that the density 7 is given as T x1 x2 x3 cpi x1 p22 x1 p3 x3 x1 That is p determines a probability density in R and p x and p2 x are conditional probability densities i e specify a probability density for each x R Figure 1 illustrates this model The reader unfamiliar with graphical models is advised to take a look at some introductory material e g 11 12 1 For such components the integral above reduces to a sum D sO N 2 Cs Figure 1 The example model graphically 1 2 Algorithms Grapham uses recently developed adaptive random walk based Markov chain Monte Carlo MCMC algorithms to produce a sequence of random variables X X2 hav ing values in the space X such that 12 LS Xi gt x n x dx I 7 LS gt J Om as n Of c
8. Grapham can be summarised as follows Let X x Rf be an initial point by default a zero vector and let 1 s gt 0 be a positive definite matrix and 0 gt 0 Define M Xj and recursively for n gt 1 Xn 1 Po S Xn 1 Mn 1 1 Nn 1 Mn Nn 1Xn 1 2 Sn4a L Nn41 Sn 141 Xn41 Mn Xn 1 Mn 3 Onti ny1 On On 1 4 where Ps x is the MCMC kernel and a 1 is the acceptance probability from the MCMC kernel The adaptation weights Nn are by default n and can be determined by defining a function para adapt_weight that receives k gt 0 as the argument and returns the value of O lt Nx 2 lt 1 The function An adapts the scaling of the proposal distribution The default function is defined as h 8 0 Oexp no Oopt 5 where ns can be defined like 1 above but using the parameter para adapt_weight_sc where Qopt is the target acceptance probability defined by para acc_opt1 and para acc_opt2 One can define a custom h by definining the function para scaling_adapt The initial value of the scaling parameter 0 is by default 2 387 d and the default value of s is the identity matrix The kernel P x is a Metropolis kernel with a symmetric proposal distribution see Section A 3 having covariance s A sample Z from P x can be simulated as follows 1 Draw a proposal random variable Y from q x 5 2 Let Z Y accept with probability a a x Y min 1
9. M algorithm of 10 which uses the covariance of the whole history of the chain in the proposal In essence dn fc where Cn Cov X1 Xn the covariance estimate of the his tory and fo is a proposal distribution with shape parameter c Adaptive scaling random walk Metropolis algorithm of 7 and 13 In this case qn fo where 8 determines the scale of the proposal distribution f The parameter 8 depends on the observed acceptance probabilities i e 6 On Q2 Qn 1 e A robust adaptive Metropolis algorithm 15 which adjusts the proposal pseudo covariance based on the directions of the proposal increments and the observed acceptance probabilities It must be noted that the theoretical validation of these adaptive MCMC algorithms is still under serious research Most of the algorithms Grapham implements have demon strated to preserve correct ergodicity i e that J J e g under some restrictive conditions on the target distribution 7 It is however believed that they work more generally The user is advised to take a look on the recent survey by Andrieu and Thoms 6 All the algorithms Grapham implements can be used block wise That is in the context of the above algorithm the steps 1 and 2 are iterated m times and each time the variable uw has only some non zero components The blocks can contain freely any components of any of the sampled variables 1 3 Technology Grapham is
10. Mn Xn Mn 7 where Y 1 is the proposed value in the n 1 th step See 6 for derivation The robust adaptive Metropolis para algorithm ram implements does not up date the scaling separately but instead of 3 the covariance factor S is updated according to the rule UUT 1 2 1 2 Sn 1 si 1 4 Moni opt Su yr where U Sy fas Xn is the normalised proposal increment see 15 for details A 2 The Initialisation Strategies There are three initialisation strategies available which can be specified in para init The default strategy greedy stands for continuous adaptation that is performed both during burn in and estimation That is 1 4 are applied all the time The safe option freeze means that adaptation is frozen after burn in and the estimation is performed with standard Metropolis kernel This means that for n gt Nburn in only the equation 1 is applied The option trad means an adaptation in the spirit of the seminal paper 10 This means that for n lt Mburn in 1 is replaced with Xn 1 Pes Xn 8 When n gt Mburn in then 1 applies It is also possible to determine a proposal density that is a mixture of two proposal densities That is 1 is replaced with Xn 1 Pn 1Pe s Xn 1 Pn 1 Pons Xn 9 18 where 0 lt py41 lt 1 is the probability of drawing from the initial proposal density This is achieved by the field para p_mi
11. One can supply the functional whose expectation over the target distribution is es timated during the simulation run In many cases e g when the functional is com plicated it is more convenient to export the simulation data see Section 7 2 and implement the functional in some more sophisticated environment such as R or Mat lab The functional is defined as a global function functional returning a lua table of numbers a vector For example suppose one is interested in computing the first and second moment of the real variable x then one could define function functional return x x 2 end Alternatively if one supplies the functional in a C library see Section 8 then the information of the functional is a given in a Lua table The fields of the table are Field Description name The name of the C function dim The dimension of the output of the function args The arguments i e the variables the functional depends on They are supplied as input arguments to the function when its value is evaluated 10 6 Simulation Parameters The simulation parameters are defined in the global table para with any collection of the following fields For detailed information on the fields see Appendix A Field Description niter Number of actual MCMC iterations default 10e3 nburn Number of additional burn in iterations default 0 nthin Thinning if nthin gt 1 then only every nthin th sample is used when c
12. ation The graphical model under consideration is determined in Grapham as the table of name model Each of the elements of this table define one node or random element The node may have a number of parents of which the conditional density of the variable depends on For example the model of Figure 1 with three nodes x1 x2 and x3 could be given as parents density SA parents density In general the fields Field dim type kind parents init_val function x return dnorm x 0 1 end x1 function x x1 return dexp x 1 x1 2 end x1 function x x1 return dnorm x x1 0 1 end describing each node are as follows Description The dimension of the node integer gt 1 scalar or vector or a two element table with integers gt 1 a matrix The Lua type of the node is one of the following number a scalar which must be of dimension one vector a Lua table of length specified by dimension In the case of a matrix a table of rows which are tables matrix Numlua matrix class custom custom structure indexed with Lua table access metamethods The kind of the node either real the default or integer in which case the feasible values of the node are the integers The list of strings with the names of the parent nodes The initial value of the node default all elements zero Field density limits Description A function returning the logar
13. for U 19
14. freedom where the last Wishart distribution is for matrices To see what the distributions look like one can see e g Wikipedia http en wikipedia org wiki Category Probability_distributions For implementation details see src grapham_math c 4 1 Constants It is possible to determine dummy nodes in the model that do not have parent variables and have constant values This can be done by defining a table const of which each element specifies a variable and the value associated to it For example const a 1 b 2 4 determines two constants real constant a and vector constant b The const table is actually just a shorthand notation for the definition of prior nodes that are instantiated with some values Note that you do not need to define all constants inside const Instead you can define any number of global variables and functions and use them freely anywhere in the model When speeding up the simulation however it is necessary to define some constants inside const 4 2 Data The instantiations i e the fixed values for some nodes in the model are given in the table data Each element of data instantiates one variable in the model to a certain value For example data x 1 2 z 3 instantiate the two dimensional vector variable x into value 1 2 and the real variable z to value 3 4 3 Repeated Blocks There ary many cases in which it is convenient to have repeated blocks in
15. grapham_read r gt data lt grapham_read out bin gt plot data Similarly in Matlab one could use the commands gt gt addpath tools gt gt data grapham_read out bin gt gt plot data z data x_2_ to read and plot the data in the file out bin 8 Speeding Up the Simulation Even though Grapham is not intended to be the fastest MCMC simulation tool there are some ways that may speed the simulation up a bit First of all instead of defining a density function that just passes the arguments to a predefined C function such as parents m v density function x m v return dnorm x m v end one may define the above as m 3 y dnorm in which case one call of Lua function is avoided as the C function dnorm is called directly with the arguments x m v Also the user may define custom densities in C An example model specification 14 dim 2 density dcircular parents c para clib clib dcircular so outfile circular bin and the file dcircular c which is compiled into shared library file dcircular so include lt stdio h gt include lt stdlib h gt include grapham_math h double dcircular const double arg const int len const int N if N 2 lea 0 2 lenf1 1 fprintf stderr dcircular Invalid arguments n exit EXIT_FAILURE const double x arg 0 double c arg 1
16. ithmic density values of the con ditional probability density p self parents Either a string of the function name or a Lua function definition The func tion is called with p 1 arguments where the first argument is the variable itself and the rest are the parent variables in the order they appear in the field parents Note the value NINF is the negative infinity which density may return out of its support A table of two elements determining the limits where the den sity is truncated The first value is the lower bound and the second the upper bound The field can be set only with uni variate distributions The values INF and NINF are acceptable or alternatively nil when there is no lower or upper limit re spectively There is a number of ready made density functions in Grapham The continuous one dimensional densities are briefly listed below Function Parameters Description dbeta a B Beta distribution with shapes amp gt 0 and B gt 0 dchi2 k y distribution with k gt 0 degrees of freedom dcauchy xo Y Cauchy Lorentz with location xo and scale y gt 0 derlang k Erlang with shape k gt 0 and rate gt 0 dexp b Exponential with scale inverse rate b gt 0 dfisher di do F distribution Fisher Snedecor with d gt 0 and d2 gt 0 degrees of freedom dgamma k 8 Gamma with shape k gt 0 and scale 0 gt 0 dgumbel uB Gumbel Fisher Tippett with location u and scale B
17. nce Lua is a full featured programming language and provides I O routines in the io module 7 2 Exporting Data In some situations it may be insufficient only to have the average of some functional of the samples It may be more convenient to store a whole set of simulated samples and work in some other environment Grapham provides two output file formats CSV and binary The CSV format is standard and read by many programs but it has the downside that the files tend to be unnecessarily large that are slow to write by Grapham and slow to read by any other program The binary file format of Grapham produces small files that are essentially just dump of the data They are fast to write and read to another program The files consist of the header line which is identical to that of the CSV file The rest of the file is just raw dump of the binary double precision floating point numbers The data can be exported from Grapham by means of two parameters para outfile and para outftm described in Section 6 There are input routines ready for the free GNU R and proprietary Mathworks Matlab The tools can be found in the directory tools As an example run Grapham with the model file example lua grapham models example lua 13 This writes a binary output file out bin Suppose that the working directory of R is the Grapham main directory The file out bin can be read to R and visualised with the commands gt source tools
18. ng variables auto matically List of initial Cholesky factors of each of the blocks An extra scaling factor that is applied to the Cholesky factors The initial scaling factor Whether to use systematic scan the default 0 or the random scan 40 Metropolis within Gibbs sampler The name of the user supplied C library 7 Data Input and Output Grapham provides simple means for data import and export 12 7 1 Importing Data In many cases one has a set of data in some format that needs to be applied to a model in Grapham If there are few data points it may be sufficient to directly define them in the model specification However if there are very many data points it is useful to be able to import the data automatically Grapham provides only one way to import data by the means of CSV comma separated values ASCII files A typical example would be the following vars y_data read_csv data csv repeat_block vars unpack y_data In this case the data is read from the CSV file data csv If there is a header line in the file consisting of the names of the variables they are stored in the Lua table vars otherwise vars is empty The data is read into y_data where y_data i is the i th column of the CSV file The Lua function unpack makes them separate arguments to the function repeat_block Although Grapham does provide only CSV importing it is possible to write input routines for any kind of data si
19. omputing the ergodic average of the functional and saved to the output file default 1 algorithm The used algorithm am aswam rbam rbaswam asm ram metropolis default am proposal The used proposal distribution norm unif laplace cauchy student default norm blocking The blocking strategy sc node full corresponding to single component blocking in which every component of every sampled node is sampled independently or node wise blocking in which every node is sampled independently and full blocking in which all the sampled variables are in one single sampling block init The initialisation strategy greedy freeze trad default greedy outfile The name of the output file Default no output is written to a file outfimt The format of the output file bin ascii default bin outvars A list of names of the variables or components to be saved in the output file outcfg The name of the file where the output configuration most im adapt_outfile adapt_outfmt portantly the Cholesky factor of estimated covariance matrix is stored By default the output configuration is not stored The name of the output file where adaptation process is stored Default no output file The format of adapt_outfile as outfmt acc_opt The optimal acceptance probability in dimension 1 default 0 44 acc_opt2 The optimal acceptance probability in dimen
20. ourse in practice Grapham produces a finite but long sequence X1 X2 Xn and outputs the estimate J which hopefully approximates sufficiently well the integral J above In particular the chain is constructed by starting it at some fixed point X x X and for n gt 2 following the recursion 1 simulate Y X 1 Un where U are independent random vectors distributed according to the symmetric probability density qn and 2 with probability O min 1 7 X _1 the proposal is accepted and Xn Yn otherwise the proposal is rejected and X X n 1 If the proposal density qn in step 1 were fixed i e qn q for all n gt 2 this construction would follow the well known Metropolis algorithm It is a well known fact that the proposal density used in the Metropolis algorithm determines the efficiency of the algorithm The main purpose of the adaptive MCMC algorithms Grapham implements is to adjust g automatically to allow efficient sim ulation of m In particular the proposal density qn above can depend on the whole history of the chain That is dn n X1 X2 Xn 1 However these proposal densities g need to be specified with care in order to main tain the consistency i e that J I and hence J J with large n The algorithms Grapham implements are described in detail in Appendix A In particular the imple mented algorithms include e A variant of the original Adaptive Metropolis A
21. sections The command grapham can be used with different command line arguments The concise usage message is shown by grapham Cesa Usage grapham v vv q e lua code model_file s The switch v results in more output and vv even more while q suppresses the output completely excluding the functional average if one is computed The switch e can be used to provide some Lua code in the command line If e is issued before the model_file argument then the Lua code is executed before executing model_file and vice versa A typical use of e would be grapham models gamma_test lua e para niter 1e6 which runs the same test as above but with a million iterations The model file s that are provided are Lua code that are run when they are read In the example file gamma_test lua there are some auxiliary constants that are computed in the first three lines theta 2 k 3 m theta k m2 theta k 2 k theta 2 These constans are visible in all what follows in the file For general introduction to Lua you can read e g Programming in Lua The first edition is available in http www lua org pil Note however that you do not have to learn Lua completely to use Grapham The examples provided in the directory models provide quite many examples how things can be done The following sections will describe how the rest of the model specification file is built up 4 Model Specific
22. sion gt 1 de fault 0 234 11 Field scaling adapt close_hook dr adapt_weight Description A user defined function for performing the acceptance rate optimisation A function to be executed after the simulation is finished If this is set it determines the positive scaling factor typically in 0 1 of the second phase of delayed rejection A function returning the adaptation weight or a non negative real number y default y 2 3 resulting in the adaptation weight sequence is 2 y This weight is used when com puting the covariance factor adapt_weight_sc Like adapt_weight but this value is used for scale adap adapt_weight p_mix seed blocks blocks_chol blocks_chol_sc blocks_scaling random_scan clib tation with algorithms with a coerced acceptance probability asm aswam rbaswam A function returning the adaptation weight or a non negative real number y default y 1 resulting in the adaptation weight sequence is 1 2 A function returning the probability of drawing from a fixed proposal component By default not fixed component is used The random seed of the pseudorandom generator Default the generator Is initialised from system time The blocks of variables as a table of tables with names of the variables in each block If this is not specified or not all the variables are in the specified blocks the selected blocking strategy determines the blocks of remaini
23. written in the C programming language uses some Netlib Fortran nu merical routines 4 and the Lua programming language 3 in model specification In addition Grapham can take advantage of the dSFMT random number generator 5 and the Numeric Lua package 8 The advantage of these technologies is that C and Fortran are standard technologies which makes it easy to compile Grapham in basically any system Even though Lua is not standard it is also written in C and also very easy to port Moreover these choices allow a relatively good efficiency in the sense of simulation speed Other software having essentially the same purpose as Grapham include BUGS 1 and JAGS 2 It is not intended that Grapham competes with the existing software Merely the purpose is to serve as a testbed for the recently developed adaptive MCMC algorithms 2 Installation There are some binary packages of Grapham available but it is always recommended to use the latest source For the binary packages Grapham is ready for use immediately 2 More specifically intended to be ISO C99 but may fail to comply any standards 3 Provided that one can use IEEE double precision floating point numbers after unpacking see Section 3 Before installation make sure that you have Lua gt 5 1 develop ment package installed Get the latest source code of Grapham from http iki fi mvihola grapham In the simplest case the following shell commands would
24. x which is a function that receives k gt 0 as an argument and returns 0 lt pg 2 lt 1 If this function returns a a small fixed value this reflects to the modification proposed by 14 By determining a sequence of probabilities 1 p2 gt p3 gt pg 0 one can have in a sense a gradual initialisation since 9 is the same as 8 for n 1 and 9 1 asn A 3 The Proposal Densities There are different proposal densities that can be used in Grapham All the proposals use a similar generation procedure Y X LU where X is the current position L is the Cholesky factor of the covariance matrix and U is a zero mean symmetrically distributed random variable with unit variance In fact all but one of the implemented proposal distributions define U U1 U4 Rf as a vector of independent and identically distributed U The possible distributions for the distributions of U determined by para proposal are norm Gaussian N 0 1 this is the default unif Uniform distribution in interval V3 V3 laplace Laplace or two tailed exponential distribution with zero mean and scale 2 In addition the choice student determines a multivariate Student distribution with one degree of freedom That is the vector U has the density f x c 1 x7x 4 where c gt 0 is a constant Observe that only the options norm and student determine a spherically sym metric distribution
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